Alternative Algorithms: Increasing Options, Reducing Errors

Article excerpt

An algorithm is "a finite, step-by-step procedure for accomplishing a task that we wish to complete" (Usiskin 1998, p. 7). Algorithms have served as a major focus of mathematics education in the United States for decades. Because school-based mathematics focuses on computation and estimation, the tasks of developing number sense, place-value understanding, and strategies for computing with algorithms remain of great importance to elementary school teachers. "The use of algorithms allows students to look at math as a process rather than as a question answer type activity...they can choose from their toolbox. Algorithms provide a comfort zone for some students and encourage students to pursue better ways as they get comfortable with them" (Mingus and Grassl 1998, p. 56).

Some students have trouble completing whole-number algorithms, whereas others may search for interesting alternatives to traditional computation methods. The options described here present opportunities for critical thinking, for a wide variety of mathematical experiences, and for increased communication and an atmosphere of high expectations. Alternatives to the customary algorithms, commonly encountered in textbooks and resource materials, are not unique but facilitate students' conceptual and skill development at their own levels of understanding and decision making. "Students skilled in using a variety of computational techniques have at their command the power and efficiency of mathematics" (Sheffield and Cruikshank 2000, p. 154).

Addition Algorithms

The right-to-left partial-sums algorithm (see fig. la) was developed in India more than 1000 years ago (Bassarear 1997). Each sum is recorded separately in the correct place. Beginning in the ones column, the 6 ones and 9 ones represent 15 ones, or 1 ten and 5 ones. The 1 is recorded in the tens place, and the 5 is recorded in the ones place. Because the 9 and 7 represent 9 tens and 7 tens, that sum is recorded as 1 hundred and 6 tens. The 4 and 8, representing 4 hundreds and 8 hundreds, add to 12 hundreds; the 1 is recorded in the thousands place and the 2, in the hundreds place.

The advantage of this method, for students who have difficulty with the traditional algorithm, is that mental regrouping is not required; all the partial sums are recorded separately before being combined. Other students find this method enjoyable because with it they can "see" where each of the numbers in the traditional procedure originates. Also, the method yields a correct sum whether the partial sums begin with the ones or the left-most column.

Left-to-right addition (see figs. lb and 1c) was often used to check the result of right-to-left addition before 1900 (Pearson 1986). These methods are based on partial sums and correct place-value alignment. To use these algorithms, one begins on the left side and combines the 4 and 8, which represent 4 hundreds and 8 hundreds. A sum of 12 hundreds, or 1200, is recorded in the correct places, representing 1 thousand and 2 hundreds, with or without recording the 0 place holders. The 9 tens and 7 tens are recorded as 16 tens, or 160, on the next line down in the correct place. Combining the 6 ones and the 9 ones results in 15 ones, which are recorded on the third line. The places from left to right are added to obtain the sum. Using graph paper aids students in recording digits in the correct places.

The advantages of the partial-sum method include the opportunity to add from left to right for those students who have difficulty with right-to-left orientation. Because primary-grade students are taught to read from left to right, they in particular may have difficulty in working algorithms from right to left. Also, the option of beginning the algorithm with the largest place value is appealing to students; this method reflects students' tendency to group larger numbers first, just as they group larger base-ten blocks before beginning to work with the unit cubes. …